# Category: Functions and mappings

Biholomorphism
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose i
Angle of parallelism
In hyperbolic geometry, the angle of parallelism , is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment leng
Differentiable vector–valued functions from Euclidean space
In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domain
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivati
Codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term rang
Glide reflection
In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operatio
Primitive recursive set function
In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They
Semilinear map
In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist"
Diffeology
In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are. The concept was first introduc
Kolmogorov–Arnold representation theorem
In real analysis and approximation theory, the Kolmogorov-Arnold representation theorem (or superposition theorem) states that every multivariate continuous function can be represented as a superposit
Equiareal map
In differential geometry, an equiareal map, sometimes called an authalic map, is a smooth map from one surface to another that preserves the areas of figures.
Map (mathematics)
In mathematics, a map is often used as a synonym for a function, but may also refer to some generalizations. Originally, this was an abbreviation of mapping, which often refers to the action of applyi
Correlation (projective geometry)
In projective geometry, a correlation is a transformation of a d-dimensional projective space that maps subspaces of dimension k to subspaces of dimension d − k − 1, reversing inclusion and preserving
Bijection, injection and surjection
In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from
Oscillation (mathematics)
In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point. As is
Transcendental function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcen
Richardson's theorem
In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, and exponential and sine functions. It was proved in 1
Local diffeomorphism
In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition o
List of set identities and relations
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It
Parity function
In Boolean algebra, a parity function is a Boolean function whose value is one if and only if the input vector has an odd number of ones. The parity function of two inputs is also known as the XOR fun
Monogenic function
A monogenic function is a complex function with a single finite derivative. More precisely, a function defined on is called monogenic at , if exists and is finite, with: Alternatively, it can be defin
Point reflection
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is sai
Swish function
The swish function is a mathematical function defined as follows: where β is either constant or a trainable parameter depending on the model. For β = 1, the function becomes equivalent to the Sigmoid
Rosenbrock function
In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It
Semi-differentiability
In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifical
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived fr
Y-intercept
In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept or vertical intercept is a point where t
Unimodality
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisf
Rigid transformation
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between eve
Periodic summation
In signal processing, any periodic function with period P can be represented by a summation of an infinite number of instances of an aperiodic function , that are offset by integer multiples of P. Thi
Reflection (mathematics)
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dime
Conway base 13 function
The Conway base 13 function is a function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem. In other words, it is a function that s
Softmax function
The softmax function, also known as softargmax or normalized exponential function, converts a vector of K real numbers into a probability distribution of K possible outcomes. It is a generalization of
Set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which
Local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If is a local h
Generalized Ozaki cost function
In economics the generalized-Ozaki cost is a general description of cost described by Shuichi Nakamura. For output y, at date t and a vector of m input prices p, the generalized-Ozaki cost, c, is
One-sided limit
In calculus, a one-sided limit refers to either one of the two limits of a function of a real variable as approaches a specified point either from the left or from the right. The limit as decreases in
Asano contraction
In complex analysis, a discipline in mathematics, and in statistical physics, the Asano contraction or Asano–Ruelle contraction is a transformation on a separately affine multivariate polynomial. It w
Piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function appl
Hypercomplex analysis
In mathematics, hypercomplex analysis is the basic extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number. The first instance is function
Second derivative
In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative measures how the rate of change of
Homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous invers
Self-concordant function
In optimization, a self-concordant function is a function for which or, equivalently, a function that, wherever , satisfies and which satisfies elsewhere. More generally, a multivariate function is se
Domain of a function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by or , where f is the function. More precisely, given a function , the domain of f is X
Rayleigh dissipation function
In physics, the Rayleigh dissipation function, named after Lord Rayleigh, is a function used to handle the effects of velocity-proportional frictional forces in Lagrangian mechanics. If the frictional
A-equivalence
In mathematics, -equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs. Let and be two manifolds, and let be two smooth map germs. We say that and are -equ
Unary function
A unary function is a function that takes one argument. A unary operator belongs to a subset of unary functions, in that its range coincides with its domain. In contrast, a unary function's domain may
List of limits
This is a list of limits for common functions such as elementary functions. In this article, the terms a, b and c are constants with respect to SM
Sammon mapping
Sammon mapping or Sammon projection is an algorithm that maps a high-dimensional space to a space of lower dimensionality (see multidimensional scaling) by trying to preserve the structure of inter-po
Tetraview
A tetraview is an attempt to graph a complex function of a complex variable, by a method invented by . A graph of a real function of a real variable is the set of ordered pairs (x,y) such that y = f(x
Injective function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, f(x1) = f(x2) implies x
Jónsson function
In mathematical set theory, an ω-Jónsson function for a set x of ordinals is a function with the property that, for any subset y of x with the same cardinality as x, the restriction of to is surjectiv
Effective domain
In convex analysis, a branch of mathematics, the effective domain is an extension of the domain of a function defined for functions that take values in the extended real number line In convex analysis
History of the function concept
The mathematical concept of a function emerged in the 17th century in connection with the development of the calculus; for example, the slope of a graph at a point was regarded as a function of the x-
List of mathematical functions
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a la
Shear mapping
In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and
Linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping
Perspectivity
In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point.
Morphism of algebraic varieties
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic va
Hubbard–Stratonovich transformation
The Hubbard–Stratonovich (HS) transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard. It is used
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object is said to be embedde
Crystal Ball function
The Crystal Ball function, named after the Crystal Ball Collaboration (hence the capitalized initial letters), is a probability density function commonly used to model various in high-energy physics.
Unfolding (functions)
In mathematics, an unfolding of a smooth real-valued function ƒ on a smooth manifold, is a certain family of functions that includes ƒ.
Jouanolou's trick
In algebraic geometry, Jouanolou's trick is a theorem that asserts, for an algebraic variety X, the existence of a surjection with affine fibers from an affine variety W to X. The variety W is therefo
Pairing function
In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number. Any pairing function can be used in set theory to prove that integers and rational
Laver function
In set theory, a Laver function (or Laver diamond, named after its inventor, Richard Laver) is a function connected with supercompact cardinals.
Vector projection
The vector projection of a vector a on (or onto) a nonzero vector b, sometimes denoted (also known as the vector component or vector resolution of a in the direction of b), is the orthogonal projectio
Locally finite operator
In mathematics, a linear operator is called locally finite if the space is the union of a family of finite-dimensional -invariant subspaces. In other words, there exists a family of linear subspaces o
Transformation (function)
In mathematics, a transformation is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f : X → X.Examples include linear transformations of vector spaces and g
Strongly unimodal
No description available.
3D projection
A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspe
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the ear
Squeeze mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapp
Vertical line test
In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y, for each unique input, x. If a vertical line int
Elasticity of a function
In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output) at point a is defined as or equivalently It is thus
Differential coefficient
In physics, the differential coefficient of a function f(x) is what is now called its derivative df(x)/dx, the (not necessarily constant) multiplicative factor or coefficient of the differential dx in
Geometric transformation
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and
Pfaffian function
In mathematics, Pfaffian functions are a certain class of functions whose derivative can be written in terms of the original function. They were originally introduced by Askold Khovanskii in the 1970s
Identity function
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. T
Translation of axes
In mathematics, a translation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the x' axis is parallel to the x axis and k
Inclusion map
In mathematics, if is a subset of then the inclusion map (also inclusion function, insertion, or canonical injection) is the function that sends each element of to treated as an element of A "hooked a
Posynomial
A posynomial, also known as a posinomial in some literature, is a function of the form where all the coordinates and coefficients are positive real numbers, and the exponents are real numbers. Posynom
Splitting lemma (functions)
In mathematics, especially in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a ne
Arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbe
Amplitwist
In mathematics, the amplitwist is a concept created by Tristan Needham in the book Visual Complex Analysis (1997) to represent the derivative of a complex function visually.
Steiner's calculus problem
Steiner's problem, asked and answered by , is the problem of finding the maximum of the function It is named after Jakob Steiner. The maximum is at , where e denotes the base of the natural logarithm.
Tapering (mathematics)
In mathematics, physics, and theoretical computer graphics, tapering is a kind of shape deformation. Just as an affine transformation, such as scaling or shearing, is a first-order model of shape defo
Iterated function
In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f : X → X with itself a certain number of times
Superfunction
In mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function f and for some variable x, the superfunction could
Discontinuous linear map
In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see li
Inversion transformation
In mathematical physics, inversion transformations are a natural extension of Poincaré transformations to include all conformal one-to-one transformations on coordinate space-time. They are less studi
Function (mathematics)
In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the fun
Rotation of axes
In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes
Surjective function
In mathematics, a surjective function (also known as surjection, or onto function) is a function f that every element y can be mapped from element x so that f(x) = y. In other words, every element of
Partial permutation
In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set Sis a bijection between two specified subsets of S. That is, it is defined by two subsets U and V o
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function f that is its own inverse, f(f(x)) = x for all x in the domain of f. Equivalently, applying f twice produces
Poincaré transformation
No description available.
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is pai
A Primer of Real Functions
A Primer of Real Functions is a revised edition of a classic Carus Monograph on the theory of functions of a real variable. It is authored by R. P. Boas, Jr and updated by his son Harold P. Boas.
Perspective (graphical)
Linear or point-projection perspective (from Latin: perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear p
Antilinear map
In mathematics, a function between two complex vector spaces is said to be antilinear or conjugate-linear if hold for all vectors and every complex number where denotes the complex conjugate of Antili
Function composition
In mathematics, function composition is an operation  ∘  that takes two functions f and g, and produces a function h = g  ∘  f such that h(x) = g(f(x)). In this operation, the function g is applied to
Antiautomorphism
No description available.
Carmichael function
In number theory, a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that holds for every integer a coprime to n. In algebraic terms,
Functional decomposition
In mathematics, functional decomposition is the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recompose
Parent function
In mathematics, a parent function is the simplest function of a family of functions that preserves the definition (or shape) of the entire family. For example, for the family of quadratic functions ha
Range of a function
In mathematics, the range of a function may refer to either of two closely related concepts: * The codomain of the function * The image of the function Given two sets X and Y, a binary relation f be
Corestriction
In mathematics, a corestriction of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a
Oblique reflection
In Euclidean geometry, oblique reflections generalize ordinary reflections by not requiring that reflection be done using perpendiculars. If two points are oblique reflections of each other, they will
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted a
Primitive recursive function
In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number
Anderson function
Anderson functions describe the projection of a magnetic dipole field in a given direction at points along an arbitrary line. They are useful in the study of magnetic anomaly detection, with historica
Zero of a function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function , is a member of the domain of such that vanishes at ; that is, the function attains the
Bell-shaped function
A bell-shaped function or simply 'bell curve' is a mathematical function having a characteristic "bell"-shaped curve. These functions are typically continuous or smooth, asymptotically approach zero f
Shekel function
The Shekel function is a multidimensional, multimodal, continuous, deterministic function commonly used as a test function for testing optimization techniques. The mathematical form of a function in d
Function application
In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be
Homography (computer vision)
In the field of computer vision, any two images of the same planar surface in space are related by a homography (assuming a pinhole camera model). This has many practical applications, such as image r
Onsager–Machlup function
The Onsager–Machlup function is a function that summarizes the dynamics of a continuous stochastic process. It is used to define a probability density for a stochastic process, and it is similar to th
Laguerre transformations
The Laguerre transformations or axial homographies are an analogue of Möbius transformations over the dual numbers. When studying these transformations, the dual numbers are often interpreted as repre
Quaternionic analysis
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real
Homomorphic secret sharing
In cryptography, homomorphic secret sharing is a type of secret sharing algorithm in which the secret is encrypted via homomorphic encryption. A homomorphism is a transformation from one algebraic str
Incomplete Bessel K function/generalized incomplete gamma function
Some mathematicians defined this type incomplete-version of Bessel function or this type generalized-version of incomplete gamma function:
Soboleva modified hyperbolic tangent
The Soboleva modified hyperbolic tangent, also known as (parametric) Soboleva modified hyperbolic tangent activation function ([P]SMHTAF), is a special S-shaped function based on the hyperbolic tangen
Propositional function
In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (x
Tak (function)
In computer science, the Tak function is a recursive function, named after (竹内郁雄). It is defined as follows: def tak( x, y, z) if y < x tak( tak(x-1, y, z), tak(y-1, z, x), tak(z-1, x, y) ) else z end
K-equivalence
In mathematics, -equivalence, or contact equivalence, is an equivalence relation between map germs. It was introduced by John Mather in his seminal work in Singularity theory in the 1960s as a technic
Signomial
A signomial is an algebraic function of one or more independent variables. It is perhaps most easily thought of as an algebraic extension of multivariable polynomials—an extension that permits exponen
High-dimensional model representation
High-dimensional model representation is a finite expansion for a given multivariable function. The expansion was first described by Ilya M. Sobol as The method, used to determine the right hand side
Alternating multilinear map
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a m
Motivic zeta function
In algebraic geometry, the motivic zeta function of a smooth algebraic variety is the formal power series Here is the -th symmetric power of , i.e., the quotient of by the action of the symmetric grou
Motor variable
In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. Will
Squeeze theorem
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is trapped between two other functions. The squeeze theorem
Similarity invariance
In linear algebra, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain. That is, is invariant under similarities if where is a matrix
Carleman matrix
In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which
Algebraic function
In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms,
Ridge function
In mathematics, a ridge function is any function that can be written as the composition of a univariate function with an affine transformation, that is: for some and .Coinage of the term 'ridge functi
Integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integ
Multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely,
Pseudoreflection
In mathematics, a pseudoreflection is an invertible linear transformation of a finite-dimensional vector space such that it is not the identity transformation, has a finite (multiplicative) order, and
Nonlocal operator
In mathematics, a nonlocal operator is a mapping which maps functions on a topological space to functions, in such a way that the value of the output function at a given point cannot be determined sol
Graph of a function
In mathematics, the graph of a function is the set of ordered pairs , where In the common case where and are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and
Partial function
In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the dom
Differentiation in Fréchet spaces
In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is